Are (non-void) self-closing tags valid in HTML5?

The W3C validator (Wikipedia) doesn’t like self-closing tags (those that end with “/>”) on non-void elements. (Void elements are those that may not ever contain any content.) Are they still valid in HTML5?

Some examples of accepted void elements:

<br />
<img src="" />
<input type="text" name="username" />

Some examples of rejected non-void elements:

<div id="myDiv" />
<span id="mySpan" />
<textarea id="someTextMessage" />

_{Note:
The W3C validator actually accepts void self-closing tags: the author originally had a problem because of a simple typo (\> instead of />); however, self-closing tags are not 100% valid in HTML5 in general, and the answers elaborate on the issue of self-closing tags across various HTML flavors.}